Integrable Functions with Equal Integrals on Sub-Sigma-Algebra are A.E. Equal

Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $\mathcal G$ be a sub-$\sigma$-algebra of $\Sigma$.

Let $f, g: X \to \overline{\R}$ be $\mathcal G$-integrable functions.

Suppose that, for all $G \in \mathcal G$:

$\displaystyle \int_G f \, \mathrm d \mu = \int_G g \, \mathrm d \mu$

Then $f = g$ $\mu$-almost everywhere.

Also see

• Measurable Functions with Equal Integrals on Sub-Sigma-Algebra are A.E. Equal

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $10.14 \ \text{(i)}$